a r X i v :h e p -p h /0107194v 1 17 J u l 2001
Parton densities and structure functions at next-to-next-to-leading order and beyond
W.L.van Neerven and A.Vogt
Instituut-Lorentz,University of Leiden,
P.O.Box 9506,2300RA Leiden,The Netherlands
Abstract.We summarize recent results on the evolution of unpolarized parton densities and deep-inelastic structure functions in massless perturbative QCD.Due to last year’s extension of the integer-moment calculations of the three-loop splitting functions,the NNLO evolution of the parton distributions can now be performed
reliably at momentum fractions x >～10
?4
,facilitating a considerably improved theoretical accuracy of their extraction from data on deep-inelastic scattering.The NNLO corrections are not dominated,at relevant values of x ,by their leading small-x terms.At large x the splitting-function series converges very rapidly,hence,employing results on the three-loop coe?cient functions,the structure functions can be analysed at N 3LO for x >10?2.The resulting values for αs do not signi?cantly change beyond NNLO,their renormalization scale dependence reaches about ±1%at N 3LO.
1.Introduction
Precise predictions for hard strong-interaction processes require transcending the standard next-to-leading order(NLO)approximation of perturbative QCD.Resumm-ations of large logarithms may be su?cient for speci?c processes,but generally full next-to-next-to-leading order(NNLO)calculations are called for.For electron–proton scattering and proton–(anti-)proton colliders this demands both partonic cross sections and parton distribution of NNLO accuracy.
The former quantities are presently available only for the structure functions in deep-inelastic scattering(DIS)[1]–which provide the backbone of our knowledge of the proton’s parton densities and are among the quantities best suited for measuringαs–and the total cross section for the Drell-Yan process[2]–which in the form of W and Z production is an excellent candidate for an accurate luminosity monitor at Tevatron and the LHC[3].The calculation of other processes like jet production at NNLO is under way,e.g.,the required two-loop two-to-two matrix elements have been computed[4] using the pioneering results[5]for the scalar double-box diagrams.See refs.[6]for recent brief overviews.Partial NNLO results(the soft-and virtual-gluon contributions) have also been obtained for Higgs production via gluon-gluon fusion in the heavy top-quark limit[7],see also ref.[8].
The three-loop splitting functions entering the NNLO evolution of the parton distributions have not been completed so far either[9].However,previous partial results[10,11]have been substantially extended by the calculation of two more Mellin moments[12].In Section2we discuss the resulting improvement[13]of our approximations of the splitting functions in x-space[14],and compare,in the extended range x>～10?4of safe applicability,the resulting approximate NNLO?avour-singlet evolution and its scale stability to the NLO results.We also brie?y re-address[15,16] the question to what extent the leading small-x contributions to the splitting functions and coe?cient functions dominate the small-x evolution.
Taking into account the fast convergence of the splitting-function series shown in Section2,the next-to-next-to-next-to-leading order(N3LO)corrections for the DIS structure functions can be e?ectively derived at x>10?2using available partial results[10,12,18]for the three-loop coe?cient functions.The e?ect of the NNLO and N3LO terms is discussed in Section3for the scaling violations of the non-singlet structure function F2and the resulting determination ofαs[19].Here we also illustrate the predictions of the principle of minimal sensitivity[20],the e?ective charge method[21] and the Pad′e summation[22]which in this case,unlike the soft-gluon resummation, seem to facilitate a reliable estimate of the corrections even beyond N3LO.
A?rst study has been performed[23]of the e?ects of the NNLO corrections(using our original approximations[14]for the three-loop splitting functions)in a global parton analysis.See refs.[24,25]for beyond-NLOαs analyses of DIS data using methods more directly based on the integer-moment results[10,12].
2.Singlet parton densities and structure functions at NNLO
We?rst illustrate our approximation procedure for the three-loop splitting functions P(2).As an example we discuss the N1f term P(2)qg,1of the gluon-quark splitting function P qg dominating the small-x evolution of the quark densities.In the
.(1)
x
The leading small-x coe?cient C1has been derived by Catani and Hautman[11].The function f smooth collects all contributions which are?nite for0≤x≤1.This regular term constitutes the mathematically complicated part of Eq.(1),involving higher transcendental functions like the harmonic polylogarithms[26].
For our improved approximations[13]we choose three or two of the large-x logarithms in Eq.(1),a one-or two-parameter smooth function(low powers or simple polynomials of x)and two of the small-x terms(x?1together with ln x or ln2x).Their coe?cients are then determined from the six even-integer Mellin moments
P(2)qg,1(N)= 10dx x N?1P(2)qg,1(x)(2) computed by Larin et al.[10]and Retey and Vermaseren[12].By varying these choices we arrive at about50approximations(see Fig.1of ref.[13]).The two functions spanning the resulting error band for most of the x-range are?nally selected as our best estimates for P(2)qg,1(x)and its residual uncertainty.
These two functions,denoted by‘A’and‘B’,are shown in Fig.1together with their (practically indistinguishable)real moments(2)for2<～N≤30and their convolutions [P(2)qg?g](x)= 1x dy y (3) with a typical gluon distribution g.In Figs.1(b)and1(c)the corresponding results for the N2f term have been supplemented for N f=4.Note that,like refs.[10,12],we use the small expansion parameter a s=αs/(4π);scaling down the ordinates by a factor 2000yields the results for an expansion inαs.The large impact of the N=10and12 moments[12]is illustrated by also showing our less accurate,but compatible original approximations[14]based on the four lowest even-integer moments[10].
Knowing the leading x?1ln x term[11]is clearly instrumental in constraining the small-x behaviour of P(2)qg,1–something not e?ciently done by a small number of N≥2 integer moments(2).However,even at x≤10?3where the non-x?1parts contribute less than10%to both approximations‘A’and‘B’,this term does not su?ciently dominate over the(so far uncalculated)subleading C0x?1contribution in Eq.(1),leaving us with a sizeable uncertainty of P(2)qg,1for x<～10?2.We will examine the dominance of the x?1ln x and x?1terms for the convolution(3),which in any case considerably smoothes out the oscillating di?erences of the approximations,at the end of this section.
After applying analogous procedures to the other three-loop splitting functions(see Figs.2and3of ref.[13])we are ready to exemplify the e?ect of the NNLO contributions
N
-40
-20
20
40
60
x
x
1010
10
101
Figure 1.(a)Exact results [10,12](points)and approximations [14,13](curves)
for the moments of the N 1
f term of the three-loop gluon-quark splittin
g function in the
x
10
10
10
10
1μr 2 / μf
20.2
0.3
0.4
101
10
0.15
0.16
0.170.18μr 2 / μf
2
0.240.26
0.28
0.3
0.3210
1
10
Figure 2.(a)Size and present approximation uncertainties of the NNLO corrections
to the scale derivatives of the singlet quark and gluon densities for the input (4)and (5)at μr =μf .(b,c)The renormalization scale dependence at NLO and NNLO for three typical values of x .
if the bands in Fig.2(a)are increased by 50%in order to account for any possible underestimate of the uncertainties.
At lower scales the splitting-function uncertainties have a larger impact,mainly due to the larger value of αs .For example,the spread corresponding to Fig.2(a)reaches
about ±4%for ˙Σand ±3%for ˙g at x =10?4and μ2f ≈3GeV 2corresponding to αs =0.3.
In view of the also enhanced NLO scale dependence,the approximate NNLO evolution represents an improvement over the NLO treatment even with inaccuracies of this size.
The electromagnetic singlet structure function F 2and its Q 2derivatives are
presented in Fig.3for the parton densities (4)at Q 2=μ2f,0≈30GeV 2
.The large NNLO corrections at very large x originate in the non-singlet part of the two-loop quark coe?cient function.Note,however,that the (positive)gluon contribution to dF 2,S /d ln Q 2at NNLO still amounts to 5%at x =0.5(40%more than at NLO),and falls below 1%only above x =0.7[14].This e?ect is large enough to jeopardize analyses applying a non-singlet formalism to the proton’s F 2in the region x >0.3.
The negative NNLO corrections to F 2at small x arise from the two-loop gluon
coe?cient function c (2)
2,g .The Q 2derivative,on the other hand,receives a +10%NNLO
correction for 10?4<～x <～10
?2;its break-up is illustrated in Fig.3(a)by the results for P (2)=0.Also for F 2and its scaling violations the inclusion of the NNLO terms leads to
1010
10
10
1
x
10
10
10
10
1
μ2/ Q
2
1.52
2.53
10
1
10
μ2/ Q
2
0.6
0.7
0.8
0.910
1
10
Figure 3.(a)The NNLO corrections for the singlet F 2and its Q 2derivatives (linear
at small x ,logarithmic at large x )for the input (4)at μ2r =μ2f ≡μ2=μ2f,0=Q 2
≈30GeV 2.(b,c)The scale dependence at NLO and NNLO for three typical values of x .
a substantial decrease of the scale uncertainties as shown in Figs.3(b,c),which facilitates more precise extractions of the parton distributions from data on these quantities.
We conclude this section by examining the dominance of the small-x terms of the NNLO splitting functions and coe?cient functions for the small-x convolutions.In Fig.4
the results for f ?g ,f =P (2)gg ,P (2)
qg and c (2)2,g obtained by keeping only the x ?1ln k x terms of f are compared,down to x =10?6,with the (for P (2)approximate)full results.The dependence on the gluon distribution g is illustrated by employing,besides our ‘steep’standard input (4),also a low-scale ‘?at’shape,xg ～x 0for x →0.
Keeping only the leading x ?1ln x terms [11]of P (2)gg and P (2)
qg does not lead to reasonable approximations as shown in Figs.4(a,b),regardless of the gluon distribution.Even for the more favourable ?at shape the o?sets amount to about a a factor of two even at x =10?6.Besides the x ?1contributions,the non-x ?1terms do not seem to be su?ciently suppressed either,at least for a steep gluon distribution.Due to the present large uncertainties [13]on the x ?1terms,however,de?nite conclusions especially for a
?at gluon distribution require the computation [9]of the exact x -dependence of P (2)
ij .On the other hand,such conclusions can be drawn already [15]for the convolutions of
the two-loop coe?cient function c (2)
2,g [1]shown in Fig.4(c).The leading x ?1term [11]does not dominate over the non-x ?1contributions at any x -values of practical interest.
x
101010101010
1
x
10
10
10
10
10
10
1
-1
01
2
10?2 (c (2) ? g) /
g
2,g N f = 4, xg = x
?0.37
(1?x)
5
x
(c)
xg = (1?x)
5
1/x full
-4
-2
0246
10-610-510-410-310-210
-11
Figure 4.The convolutions of the leading (and,for P (2),subleading)small-x terms of (a,b)the NNLO splitting functions P (2)gg and P (2)
qg and (c)the NNLO
coe?cient function c (2)
2,g with two typical ‘steep’(upper row)and ‘?at’(lower row)gluon distributions,compared to the respective (for P (2)approximate)full results.
3.Non-singlet structure functions at NNLO and beyond
The scaling violations of the non-singlet structure functions F a,NS ,a =1,2,3,can be conveniently discussed in terms of the physical evolution kernels K a,NS ,
d
N
-0.06
-0.04
-0.02
N
N
Figure 5.The PMS,ECH and Pad′e estimates of (a)the NNLO,(b)the N 3LO,and (c)the N 4LO contributions to the N -space evolution kernel for 1
x
-0.6
-0.4
-0.2
0.2
0.4
0.60.8
1
x
-0.1
-0.08
-0.06
-0.04
-0.02
0.02
x
0.02
0.04
0.06
0.08
Figure 6.(a)The perturbative expansion of the logarithmic Q 2derivative of F 2,NS
for the input (7)for N f =4and μ2r =Q 2.(b)The N n
LO contributions,n =1...4,to the results shown in (a),and (c)their renormalization scale uncertainties.
The e?ect of the higher-order terms (using the Mellin inverse of Fig.5(c)at N 4LO,see Fig.12of ref.[19]for the Pad′e estimates of the corrections beyond N 4LO)is exempli?ed in Fig.6for the logarithmic Q 2derivative of
F 2,NS (x,Q 20≈30GeV 2)=x 0.5(1?x )3
,
αs (Q 20)=0.2.
(7)
Also shown are the renormalization-scale uncertainties estimated using the conventional
interval 1
Acknowledgment
This work has been supported by the European Community TMR research network ‘QCD and particle structure’under contract No.FMRX–CT98–0194.
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